- #1

- 2

- 0

**Help Please!!!!**

Calculate the first four Picard Iterates of the equation y' - y = x^2 with the condition y(0) = -1

and it was given that y'(x) = x^2 +y and y(0)= -1

Need a little help with this question... Not sure what to do.

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter brit123
- Start date

- #1

- 2

- 0

Calculate the first four Picard Iterates of the equation y' - y = x^2 with the condition y(0) = -1

and it was given that y'(x) = x^2 +y and y(0)= -1

Need a little help with this question... Not sure what to do.

- #2

- 173

- 0

The definition of Picard iterates is

[tex]\phi_0=y(x_0),[/tex]

[tex]\phi_{n+1}(x)=\phi_0+\int_{x_0}^x f(s,\phi_n(s))ds,[/tex]

where [itex]f[/itex] is given by

[tex]y'(x)=f(x,y(x)).[/itex]

In your case, [itex]f(x,y(x))=x^2+y[/itex], [itex]\phi_0=y(0)=-1[/itex], and

[tex]\phi_1=-1+\int_0^x (s^2+\phi_0)ds=-1+\int_0^x (s^2-1)ds.[/tex]

What is [itex]\phi_2[/itex]? and [itex]\phi_n[/itex]?

[tex]\phi_0=y(x_0),[/tex]

[tex]\phi_{n+1}(x)=\phi_0+\int_{x_0}^x f(s,\phi_n(s))ds,[/tex]

where [itex]f[/itex] is given by

[tex]y'(x)=f(x,y(x)).[/itex]

In your case, [itex]f(x,y(x))=x^2+y[/itex], [itex]\phi_0=y(0)=-1[/itex], and

[tex]\phi_1=-1+\int_0^x (s^2+\phi_0)ds=-1+\int_0^x (s^2-1)ds.[/tex]

What is [itex]\phi_2[/itex]? and [itex]\phi_n[/itex]?

Last edited:

Share: